In connection with a number of features, as well as in view of the particular importance of the question of the potential energy of the forces of universal gravitation, it is necessary to consider separately and in more detail.

We come across the first feature when choosing the origin of the potential energies. In practice, it is necessary to calculate the movements of a given (test) body under the action of the forces of universal gravitation created by other bodies of different masses and sizes.

Let us assume that we have agreed to consider the potential energy equal to zero in such a position in which the bodies touch. Let the test body A, when interacting separately with balls of the same mass, but different radii, is initially removed from the centers of the balls at the same distance (Fig. 5.28). It is easy to see that when the body A moves until it touches the surfaces of the bodies, the gravitational forces will do different work. This means that we must consider the potential energies of the systems to be different for the same relative initial positions of the bodies.

It will be especially difficult to compare these energies with each other in cases when interactions and movements of three or more bodies are considered. Therefore, for the forces of universal gravitation, such an initial level of reference of potential energies is sought, which could be the same, common, for all bodies in the Universe. Such a general zero level of potential energy of the forces of universal gravitation agreed to consider the level corresponding to the location of bodies at infinitely large distances from each other. As can be seen from the law of universal gravitation, the forces of universal gravitation themselves also vanish at infinity.

With this choice of the origin of the energy count, an unusual situation is created with the determination of the values ​​of the potential energies and carrying out all the calculations.

In the cases of gravity (Fig. 5.29, a) and elasticity (Fig. 5.29, b), the internal forces of the system tend to bring the bodies to a zero level. As the bodies approach the zero level, the potential energy of the system decreases. The lowest potential energy of the system really corresponds to the zero level.

This means that for all other positions of the bodies, the potential energy of the system is positive.

In the case of the forces of universal gravitation and the choice of zero energy at infinity, everything happens the other way around. The internal forces of the system tend to take the bodies away from the zero level (Fig. 5.30). They do positive work when bodies move away from the zero level, that is, when bodies come closer. At any finite distances between the bodies, the potential energy of the system is less than at In other words, the zero level (at corresponds to the greatest potential energy. This means that at all other positions of the bodies, the potential energy of the system is negative.

In § 96 it was found that the work of the forces of universal gravitation during the transfer of a body from infinity to a distance is equal to

Therefore, the potential energy of the forces of universal gravitation must be considered equal

This formula expresses another feature of the potential energy of the forces of universal gravitation - the relatively complex nature of the dependence of this energy on the distance between bodies.

In fig. 5.31 shows a graph of dependence on for the case of attraction of bodies by the Earth. This graph looks like an isosceles hyperbola. Near the surface of the Earth, the energy changes relatively strongly, but already at a distance of several tens of Earth radii, the energy becomes close to zero and begins to change very slowly.

Any body near the Earth's surface is in a kind of "potential hole". Whenever it is necessary to free the body from the forces of gravity, special efforts must be made in order to "pull" the body out of this potential hole.

In the same way, all other celestial bodies create such potential holes around them - traps that capture and hold all not very fast moving bodies.

Knowledge of the nature of dependence on can significantly simplify the solution of a number of important practical problems. For example, you need to send a spaceship to Mars, Venus or any other planet in the solar system. It is necessary to determine what speed should be imparted to the ship when it is launched from the surface of the Earth.

In order to send a ship to other planets, it must be removed from the sphere of action of the forces of gravity. In other words, you need to raise its potential energy to zero. This becomes possible if the ship is informed of such kinetic energy that it can perform work against the forces of gravity, equal to where the mass of the ship,

mass and radius of the globe.

From Newton's second law it follows that (§ 92)

But since the ship's speed before launch is zero, it can be written simply:

where is the speed reported to the ship at launch. Substituting the value for A, we get

For exclusion, we will use, as was already done in § 96, two expressions for the force of gravity on the surface of the Earth:

Hence - Substituting this value into the equation of Newton's second law, we obtain

The speed required to remove the body from the sphere of action of the forces of gravity is called the second cosmic speed.

In the same way, you can formulate and solve the problem of sending a ship to distant stars. To solve such a problem, it is already necessary to determine the conditions under which the ship will be removed from the sphere of action of the forces of attraction of the Sun. Repeating all the reasoning that was carried out in the previous problem, you can get the same expression for the speed imparted to the ship at launch:

Here a is the normal acceleration that the Sun imparts to the Earth and which can be calculated from the nature of the Earth's motion in its orbit around the Sun; radius of the earth's orbit. Of course, in this case it means the speed of the ship relative to the Sun. The speed required to take the ship out of the solar system is called the third cosmic speed.

The method we have considered for choosing the origin of the potential energy is also used in calculating the electrical interactions of bodies. The concept of potential wells is also widely used in modern electronics, solid state theory, atomic theory, and nuclear physics.

Speed

Acceleration

Called tangential acceleration magnitude

Are called tangential acceleration characterizing the change in speed along direction

Then

V. Geisenberg,

Dynamics

Force

Inertial frames of reference

Frame of reference

Inertia

Inertia

Newton's laws

Newton's th law.

inertial systems

Newton's th law.

3rd Newton's Law:

4) The system of material points. Internal and external forces. The momentum of a material point and the momentum of a system of material points. Impulse conservation law. Conditions for its applicability of the law of conservation of momentum.

Material points system

Internal forces:

External forces:

The system is called closed system if on the bodies of the system external forces do not act.

Material point momentum

Impulse conservation law:

If and wherein hence

Galileo's transformations, the principle regarding Galileo

center of mass .

Where is the mass of i - that particle

Center mass speed

6)

Work in mechanics

)

potential .

non-potential.

The first includes

Complex: called kinetic energy.

Then Where are the external forces

Keene. the energy of the system of bodies

Potential energy

Equation of moments

The derivative of the angular momentum of a material point with respect to the fixed axis in time is equal to the moment of the force acting on the point relative to the same axis.

The sum of all internal forces relative to any point is equal to zero. That's why

Thermal coefficient of performance (COP) of the cycle Thermal engine.

The measure of the efficiency of converting the heat supplied by the amount to the working fluid into the work of the heat engine on external bodies is efficiency heat machine

Thermodynamic CRD:

Heat machine: when converting thermal energy into mechanical work. The main element of a heat engine is the work of bodies.

Energy cycle

Refrigeration machine.

26) Carnot cycle, efficiency of the Carnot cycle. Second started thermodynamics. Its various
wording.

Carnot cycle: this cycle consists of two isothermal processes and two adiabats.

1-2: Isothermal process of gas expansion at heater temperature T 1 and supplies heat.

2-3: Adiabatic gas expansion process with the temperature decreasing from T 1 to T 2.

3-4: Isothermal process of gas compression in this case heat is removed and the temperature is equal to T 2

4-1: The adiabatic process of gas compression while the gas temperature develops from the refrigerator to the heater.

Affects for the Carnot cycle, the general efficiency factor is the manufacturer

In a theoretical sense, this cycle will maximum among possibly Efficiency for all cycles operating between temperatures T 1 and T 2.

Carnot's theory: The useful power factor of the Karnot heat cycle does not depend on the type of worker and the structure of the machine itself. And only determined by the temperatures T n and T x

Second started thermodynamics

The second law of thermodynamics determines the direction of flow of heat engines. It is not possible to construct a thermodynamic cycle operating a heat engine without a refrigerator. In this cycle, the energy of the system will see….

In this case, the efficiency

Its various formulations.

1) The first formulation: "Thomson"

A process is impossible, the only result of which is the performance of work by cooling one body.

2) The second formulation: "Clausis"

A process is impossible, the only result of which is the transfer of heat from a cold body to a hot one.

27) Entropy is a function of the state of a thermodynamic system. Calculation of entropy change in ideal gas processes. Clausius inequality. The main property of entropy (formulation of the second law of thermodynamics through entropy). The statistical meaning of the second principle.

Clausius inequality

The initial condition is the second law of thermodynamics, Clausius obtained the relation

The equal sign of the reversible cycle and process, respectively.

Most likely

The speed of molecules, respectively, the maximum value of the distribution function is called the most probable probability.

Einstein's postulates

1) Einstein's principle of relativity: all physical laws are the same in all inertial reference systems, and therefore they must be formulated in a form that is invariant with respect to coordinate transformations reflecting the transition from one IFR to another.

2)
The principle of constancy of the speed of light: there is a limiting speed of propagation of interactions, the value of which in all IFR is the same and equal to the speed of an electromagnetic wave in vacuum and does not depend on the direction of its propagation, not on the movement of the source and receiver.

Consequences from the Lorentz transformations

Lorentzian length contraction

Consider a rod located along the OX 'axis of the system (X', Y ', Z') and fixed relative to this coordinate system. Own length of the bar called the value, that is, the length measured in the reference system (X, Y, Z) will be

Consequently, the observer in the system (X, Y, Z) finds that the line of the moving rod is one times less than its own length.

34) Relativistic dynamics. Newton's second law applied to large
speeds. Relativistic energy. The connection between mass and energy.

Relativistic dynamics

The connection between the momentum of a particle and its velocity is now given by

Relativistic energy

A particle at rest has energy

This quantity is called the rest energy of the particle. The kinetic energy is obviously equal to

The connection between mass and energy

Total energy

Insofar as

Speed

Acceleration

Along the tangent trajectory at its given point Þ a t = eRsin90 o = eR

Called tangential acceleration characterizing the change in speed along magnitude

Along a normal trajectory at a given point

Are called tangential acceleration characterizing the change in speed along direction

Then

The limits of applicability of the classical way of describing the movement of a point:

All of the above refers to the classical way of describing the movement of a point. In the case of a nonclassical consideration of the motion of microparticles, the concept of the trajectory of their motion does not exist, but one can speak of the probability of finding a particle in one or another region of space. For a microparticle, it is impossible to simultaneously specify the exact values ​​of the coordinate and velocity. In quantum mechanics, there is uncertainty relation

V. Geisenberg, where h = 1.05 ∙ 10 -34 J ∙ s (Planck's constant), which determines the errors of simultaneous measurement of the coordinate and momentum

3) Dynamics of a material point. Weight. Force. Inertial frames of reference. Newton's laws.

Dynamics- this is a branch of physics, it studies the movement of bodies in connection with the reasons that return one or the force of the nature of the movement

Mass is a physical quantity that corresponds to the ability of physical bodies to maintain their translational motion (inertia), as well as characterizing the amount of matter

Force- a measure of sharing between bodies.

Inertial frames of reference: There are such reference systems of relative, in which the body is at rest (moves equal to the line) until other bodies act on it.

Frame of reference- inertial: any other motion relative to heliocentrism is uniform and straight, is also inertial.

Inertia- This is a phenomenon associated with the ability of bodies to maintain their speed.

Inertia- the ability of a material body to reduce its speed. The more inert the body is, the “harder” it is to change it v. The quantitative measure of inertia is body mass, as a measure of body inertia.

Newton's laws

Newton's th law.

There are such frames of reference called inertial systems, in which the material point is in a state of rest or uniform linear motion until the impact from other bodies brings it out of this state.

Newton's th law.

The force acting on a body is equal to the product of the body's mass by the acceleration imparted by this force.

3rd Newton's Law: the forces with which two m. points act on each other in IFR are always equal in magnitude and directed in opposite directions along the straight line connecting these points.

1) If body A is acted upon by a force from body B, then body B is acted upon by force A. These forces F 12 and F 21 have the same physical nature

2) The force interact between the bodies, does not depend on the speed of the bodies

Material points system: it is such a system contained by points that are rigidly connected to each other.

Internal forces: The forces of interaction between the points of the system are called internal forces

External forces: Forces interact on the points of the system from the side of bodies that are not included in the system are called external forces.

The system is called closed system if on the bodies of the system external forces do not act.

Material point momentum is called the product of mass by the speed of a point Material point system momentum: The momentum of a system of material points is equal to the product of the mass of the system by the speed of movement of the center of masses.

Impulse conservation law: For a closed system, bodies interact, the total impulse of the system remains unchanged, regardless of any interacting bodies with each other

Conditions for its applicability of the law of conservation of momentum: The momentum conservation law can be used under closed conditions, even if the system is not closed.

If and wherein hence

The law of conservation of momentum also works in a micrometer, when classical mechanics does not work, the momentum is preserved.

Galileo's transformations, the principle regarding Galileo

Let we have 2 inertial reference frames, one of which moves relative to the second, with a constant speed v o. Then, in accordance with the Galileo transformation, the acceleration of the body in both frames of reference will be the same.

1) The uniform and rectilinear movement of the system does not affect the course of the mechanical processes taking place in them.

2) We put all inertial systems as properties equivalent to each other.

3) No mechanical experiments inside the system can establish the system at rest or it moves uniformly or rectilinearly.

The relativity of mechanical motion and the sameness of the laws of mechanics in different inertial frames of reference are called Galileo's principle of relativity

5) The system of material points. The center of mass of the system of material points. A theorem on the motion of the center of mass of a system of material points.

Any body can be represented as a collection of material points.

Let it have a system of material points with masses m 1, m 2, ..., m i, the positions of which relative to the inertial reference frame are characterized by vectors, respectively, then, by definition, the position center of mass system of material points is determined by the expression: .

Where is the mass of i - that particle

- characterizes the position of this particle relative to a given coordinate system,

- characterizes the position of the center of mass of the system relative to the same coordinate system.

Center mass speed

The momentum of a system of material points is equal to the product of the mass of the system by the speed of movement of the center of masses.

If that is the system, we say that the system as a center is at rest.

1) The center of mass of the system of motion is as if the entire mass of the system was concentrated in the center of mass, and all forces act on the bodies of the system were applied to the center of mass.

2) The acceleration of the center of mass does not depend on the points of application of forces acting on the body of the system.

3) If (acceleration = 0) then the impulse of the system will not change.

6) Work in mechanics. The concept of a field of forces. Potential and non-potential forces. The criterion for the potentiality of the field forces.

Work in mechanics: The work of the force F on an element of displacement is called the dot product

Work is an algebraic quantity ( )

The concept of the field of forces: If a certain force acts on the body at each material point of space, then they say that the body is in the field of forces.

Potential and non-potential forces, criterion for the potentiality of field forces:

From the point of view of production work, it will mark potential and non-potential bodies. Forces, for everyone:

1) Work does not depend on the shape of the trajectory, but depends only on the initial and final position of the body.

2) Work, which is equal to zero along closed trajectories, are called potentials.

The forces are convenient to these conditions is called potential .

The forces are not convenient to these conditions is called non-potential.

The first includes and only by the shear force of friction is it non-potential.

7) Kinetic energy of a material point, a system of material points. The theorem on the change in kinetic energy.

Complex: called kinetic energy.

Then Where are the external forces

The theorem on the change in kinetic energy: change kin. the energy of a point is equal to the algebraic sum of the work of all forces applied to it.

If several external forces simultaneously act on the body, then the change in the critical energy is equal to the "allebraic work" of all forces acting on the body: this is the kinetic kinetic theorem formula.

Keene. the energy of the system of bodies called amount of kin. energies of all bodies included in this system.

8) Potential energy. Change in potential energy. Potential energy of gravitational interaction and elastic deformation.

Potential energy- physical wilichina, the change of which is equal to the work of the potential force of the system taken with the “-” sign.

We introduce some function W p, which is the potential energy f (x, y, z), which we define as follows

The “-” sign indicates that when this potential force is doing work, the potential energy decreases.

Change in potential energy of the system bodies, between which only potential forces act, is equal to the work of these forces taken with the opposite sign during the transition of the system from one state to another.

Potential energy of gravitational interaction and elastic deformation.

1) Gravitational force

2) The work of the force of elasticity

9) Differential relationship between potential force and potential energy. Scalar field gradient.

Let the movement only along the x-axis

Similarly, let the movement only be along the y or z axis, we got

The sign “-” in the formula shows that the force is always directed towards the potential energy, but the opposite is the gradient of W p.

The geometric meaning of points with the same potential energy value is called the equipotential surface.

10) The law of conservation of energy. Absolutely non-elastic and absolutely elastic central balls impact.

The change in the mechanical energy of the system is equal to the sum of the work of all non-potential forces, internal and external.

*) Mechanical energy conservation law: The mechanical energy of the system is conserved if the work of all non-potential forces (both internal and external) is equal to zero.

In this case, it is possible to merge the transition of potential energy into kinetic energy and vice versa, the total energy is constant:

*)General physical law of conservation of energy: Energy is neither created nor destroyed, it either passes from the first type to another state.

Gravitational energy

Gravitational energy - potential energy system of bodies (particles) due to their mutual gravitation.

Gravitationally coupled system- a system in which gravitational energy is greater than the sum of all other types of energies (in addition to rest energy).

The generally accepted scale, according to which, for any system of bodies located at finite distances, the gravitational energy is negative, but for infinitely remote, that is, for gravitationally of non-interacting bodies, the gravitational energy is zero... The total energy of the system, equal to the sum of the gravitational and kinetic energy, is constant. For an isolated system, the gravitational energy is communication energy... Systems with positive total energy cannot be stationary.

V classical mechanics

For two gravitating point bodies with by the masses M and m gravitational energy is equal to:

This result is obtained from Newton's law of gravity, provided that for infinitely distant bodies the gravitational energy is equal to 0. Expression for gravitational strength has the form

On the other hand, according to the definition of potential energy:

The constant in this expression can be chosen arbitrarily. It is usually chosen equal to zero, so that when r tends to infinity, tends to zero.

The same result is true for a small body located near the surface of a large one. In this case, R can be considered equal, where is the radius of a body with mass M, and h is the distance from the center of gravity of a body with mass m to the surface of a body with mass M.

On the surface of the body M we have:

If the dimensions of the body are much larger than the dimensions of the body, then the gravitational energy formula can be rewritten as follows:

where the quantity is called the acceleration of gravity. In this case, the term does not depend on the height of the body lifting above the surface and can be excluded from the expression by choosing the appropriate constant. Thus, for a small body located on the surface of a large body, the following formula is valid

In particular, this formula is used to calculate the potential energy of bodies located near the Earth's surface.

V General relativity

V general relativity along with the classical negative component of the gravitational binding energy, a positive component appears due to gravitational radiation, that is, the total energy of the gravitating system decreases in time due to such radiation.

see also

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See what "Gravitational energy" is in other dictionaries:

The potential energy of bodies due to their gravitational interaction. The term gravitational energy is widely used in astrophysics. The gravitational energy of any massive body (stars, clouds of interstellar gas), consisting of ... ... Big Encyclopedic Dictionary

The potential energy of bodies due to their gravitational interaction. The gravitational energy of a stable space object (stars, clouds of interstellar gas, star clusters) in absolute value is twice the average kinetic energy ... ... encyclopedic Dictionary

gravitational energy

gravitational energy- gravitacinė energija statusas T sritis fizika atitikmenys: angl. gravitational energy vok. Gravitationsenergie, f rus. gravitational energy, f pranc. énergie de gravitation, f; énergie gravifique, f… Fizikos terminų žodynas

Potential energy of bodies, due to their gravity. interaction. G. e. sustainable space. object (stars, clouds of interstellar gas, star clusters) in abs. the value is twice the average. kinetic energies of its constituent particles (bodies; this ... ... Natural science. encyclopedic Dictionary

- (for a given state of the system) the difference between the total energy of the bound state of a system of bodies or particles and the energy of the state in which these bodies or particles are infinitely distant from each other and are at rest: where ... ... Wikipedia

This term has other meanings, see Energy (meanings). Energy, Dimension ... Wikipedia

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- (Greek energeia, from energos active, strong). Perseverance, found in the pursuit of a goal, the ability of a higher tension of forces, in conjunction with a strong will. Dictionary of foreign words included in the Russian language. Chudinov A.N., ... ... Dictionary of foreign words of the Russian language

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If only conservative forces operate in the system, then the concept can be introduced potential energy. Let the body mass m finds-

in the gravitational field of the Earth, the mass of which M... The force of interaction between them is determined by the law of gravity

F(r) = G Mm,

where G= 6.6745 (8) × 10–11 m3 / (kg × s2) - gravitational constant; r is the distance between their centers of mass. Substituting the expression for the gravitational force into formula (3.33), we find its work when the body passes from a point with a radius vector r 1 to a point with a radius vector r 2

r 2 dr

= GMm⎜⎝r

1 r 1 r 1 2 2 1

We represent relation (3.34) as the difference between the values

A 12 = U(r 1) – U(r 2), (3.35)

U(r) = -G Mm+ C

for different distances r 1 and r 2. In the last formula C is an arbitrary constant.

If the body gets close to the Earth, which is considered to be motionless, then r 2 < r 1, 1/ r 2 – 1/ r 1> 0 and A 12 > 0, U(r 1) > U(r 2). In this case, the force of gravity does a positive job. The body passes from some initial state, which is characterized by the value U(r 1) function (3.36), in the final, with a smaller value U(r 2).

If the body moves away from the Earth, then r 2 > r 1, 1/ r 2 – 1/ r 1 < 0 и A 12 < 0,

U(r 1) < U(r 2), i.e. the force of gravity does negative work.

Function U= U(r) is a mathematical expression of the ability of gravitational forces acting in the system to perform work and, according to the above definition, represents potential energy.

Note that the potential energy is due to the mutual gravity of bodies and is a characteristic of a system of bodies, not one body. However, when considering two or more bodies, one of them (usually the Earth) is considered stationary, while the others move relative to it. Therefore, one often speaks of the potential energy of precisely these bodies in the field of forces of a stationary body.

Since in problems of mechanics it is not the value of potential energy that is of interest, but its change, the value of potential energy can be counted from any initial level. The latter determines the value of the constant in formula (3.36).

U(r) = -G Mm.

Let the zero level of potential energy correspond to the surface of the Earth, i.e. U(R) = 0, where R Is the radius of the Earth. Let us write down the formula (3.36) for the potential energy when the body is at a height h over its surface in the following form

U(R+ h) = -G Mm

R+ h

+ C. (3.37)

Assuming in the last formula h= 0, we have

U(R) = -G Mm+ C.

From here we find the value of the constant C in formulas (3.36, 3.37)

C= -G Mm.

After substitution of the value of the constant C into formula (3.37), we have

U(R+ h) = -G Mm+ G Mm= GMm⎛- 1

1 ⎞= G Mm h.

R+ h R

⎝⎜ R+ h R⎟⎠ R(R+ h)

We rewrite this formula as

U(R+ h) = mgh h,

where gh

R(R+ h)

Acceleration of free fall of a body at height

h above the surface of the Earth.

Approaching h« R we obtain the well-known expression for the potential energy if the body is at a low altitude h above the surface of the earth

Where g= G M

U(h) = mgh, (3.38)

Free fall acceleration of a body near the Earth.

A more convenient notation is adopted in expression (3.38): U(R+ h) = U(h). It can be seen from it that the potential energy is equal to the work performed by the gravitational force when the body moves from a height h above

Earth on its surface, corresponding to the zero level of potential energy. The latter serves as the basis for considering expression (3.38) as the potential energy of a body above the Earth's surface, talking about the potential energy of a body and excluding the second body, the Earth, from consideration.

Let the body mass m is on the surface of the Earth. In order for it to be on top h above this surface, an external force must be applied to the body, oppositely directed to the force of gravity and infinitely little different from it in modulus. The work that an external force will do is determined by the following relationship:

R+ h

R+ h dr

⎡1 ⎤R+ h